Problem 91 - Intermediate Algebra

Question

Perform the multiplications. All variables represent positive real numbers. See Example \(9 .\) $$ x^{3 / 5}\left(x^{7 / 5}-x^{-3 / 5}+1\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^2 + x^{3/5} - 1 \).

Step 1: Distribute the Outer Term

The expression given is \( x^{3/5}(x^{7/5} - x^{-3/5} + 1) \). The first step is to distribute \( x^{3/5} \) to each term inside the parentheses. This results in:- \( x^{3/5} \cdot x^{7/5} \)- \( x^{3/5} \cdot (-x^{-3/5}) \)- \( x^{3/5} \cdot 1 \)

Step 2: Apply the Product of Powers Rule

According to the product of powers rule, \( x^a \cdot x^b = x^{a+b} \). Apply this rule to each product:- \( x^{3/5} \cdot x^{7/5} = x^{(3/5 + 7/5)} = x^{10/5} = x^2 \)- \( x^{3/5} \cdot (-x^{-3/5}) = -x^{(3/5 - 3/5)} = -x^0 = -1 \)- \( x^{3/5} \cdot 1 = x^{3/5} \) (note that multiplying by 1 does not change the expression)

Step 3: Combine the Results

Combine the results from the previous step to form the final expression:\[ x^2 - 1 + x^{3/5} \]

Step 4: Simplify the Expression

Ensure that the expression is simplified. In this context, the expression \( x^2 + x^{3/5} - 1 \) is as simplified as it can be with respect to the powers of \( x \). Thus no further simplification is needed.

Key Concepts

Exponent Rules

Exponent rules are foundational in algebra, especially when dealing with expressions involving powers. One key rule is the **Product of Powers Rule**, which states that when you multiply two expressions with the same base, you simply add their exponents. For instance, if you have an expression like \(x^a \cdot x^b\), it simplifies to \(x^{a+b}\).
This rule is incredibly useful, as it allows you to combine and simplify expressions easily.

• **Example:** \(x^{3/5} \cdot x^{7/5} = x^{(3/5+7/5)} = x^{10/5} = x^2\). Here, you add the exponents \(3/5\) and \(7/5\) to get \(10/5\) which simplifies to 2.
• Remember that any number raised to the power of zero, like \(x^0\), is always 1, as seen when simplifying \(-x^{0}\) to \(-1\).

By mastering these rules, you'll find algebraic manipulations become much more manageable.

Polynomials

Polynomials are expressions that include variables raised to whole number exponents, combined using addition or subtraction. In this exercise, we worked with a polynomial before distributing and simplifying it.
The expression provided was \(x^{3/5}(x^{7/5} - x^{-3/5} + 1)\), and it's a polynomial that includes various terms, each having different powers of \(x\).

• Understanding how to handle each term is crucial—observe how distribution multiplies each term inside the parenthesis by the term outside.
• This results in a new polynomial where each term reflects the combined powers of \(x\).

Polynomials can have any number and degree of terms, and understanding how to simplify them is key to mastering algebra. Though it might appear complex initially, breaking down each step leads you to the solution efficiently.

Simplifying Expressions

Simplifying expressions is about making algebraic expressions as straightforward as possible. This often involves performing arithmetic operations and applying rules like those for exponents. In the given exercise, simplifying consisted primarily of distribution and employing exponent rules:

• **Distribution**: You multiply the term outside the parentheses by each term inside, which increases clarity in complex expressions.
• **Combining Like Terms**: Once each piece is individually simplified, you can combine them to achieve the simplest form of the expression.

The goal in simplifying is to get to a form that is easy to read and interpret, like ending with \(x^2 + x^{3/5} - 1\).
Achieving this helps in understanding the behavior of expressions and aids in solving algebraic equations efficiently.